Asymptotics of the spectrum of the mixed boundary value problem for the Laplace operator in a thin spindle-shaped domain

The asymptotics is examined for solutions to the spectral problem Laplace operator in a d d -dimensional thin, of diameter alttext="upper O left-parenthesis h right-parenthesis"> O ( h stretchy="false">) encoding="application/x-tex">O(h) , spindle-shaped domain alttext="normal upper Omega Superscript h"> Ω encoding="application/x-tex">\Omega ^h with Dirichlet condition on small, size alttext="h much-less-than 1"> ≪ 1 encoding="application/x-tex">h\ll 1 terminal zones Gamma Subscript plus-or-minus mathvariant="normal">Γ<!-- Γ <mml:mo>±<!-- ± </mml:msubsup> encoding="application/x-tex">\Gamma ^h_\pm and Neumann remaining part boundary alttext="partial-differential normal mathvariant="normal">∂<!-- ∂ encoding="application/x-tex">\partial \Omega . In limit as right-arrow plus 0"> stretchy="false">→<!-- → <mml:mo>+ 0 encoding="application/x-tex">h\rightarrow +0 an ordinary differential equation axis alttext="left-parenthesis negative 1 comma right-parenthesis contains-as-member z"> −<!-- − <mml:mo>, ∋<!-- ∋ <mml:mi>z encoding="application/x-tex">(-1,1)\ni z spindle arises coefficient degenerating at points alttext="z equals = encoding="application/x-tex">z=\pm moreover, without any because requirement boundedness eigenfunctions makes well-posed. Error estimates are derived one-dimensional model but case alttext="d 3"> 3 encoding="application/x-tex">d=3 it necessary construct layers near sets 2"> 2 encoding="application/x-tex">d=2 deal selfadjoint extensions operator. extension parameters depend linearly alttext="ln ln ⁡<!-- ⁡ encoding="application/x-tex">\ln h so that its eigenvalues analytic functions variable alttext="1 slash StartAbsoluteValue ln EndAbsoluteValue"> / stretchy="false">| encoding="application/x-tex">1/|\ln h| As result, all dimensions gets power-law accuracy delta Super d Baseline δ<!-- δ </mml:msub> encoding="application/x-tex">O(h^{\delta _d}) exponent alttext="delta greater-than &gt; encoding="application/x-tex">\delta _d&gt;0 First (the smallest) eigenvalues, positive null encoding="application/x-tex">(-1,1) require individual treatment. Also, infinite asymptotic series discussed, well static (without parameter) related shapes thin domains.